A conjecture of Merca on nonnegativity of theta series (Q6659399)

This paper addresses and significantly advances a conjecture of \textit{M. Merca} [Exp. Math. 30, No. 3, 364--371 (2021; Zbl 1482.05017)] regarding the nonnegativity of certain theta series. The conjecture, which itself refines an earlier proposal by \textit{G. E. Andrews} and \textit{M. Merca} [J. Comb. Theory, Ser. A 119, No. 8, 1639--1643 (2012;Zbl 1246.05014)] on truncated pentagonal number series, is a topic of interest in the interplay between partition theory and \(q\)-series.\N\NThe authors begin by focusing on two special cases of Merca's conjecture, providing rigorous refinements that not only validate the conjecture in these instances but also enrich its theoretical framework. By employing techniques from analytic number theory and \(q\)-series, the paper establishes several general nonnegativity results for theta series. These results solidify the foundations of Merca's conjecture and provide insights into the underlying combinatorial structures.\N\NOf particular significance are the applications presented in the latter part of the paper. The authors connect their results to two well-known partition statistics, demonstrating positivity results that reinforce the broader implications of their findings. These applications highlight the relevance of the conjecture in the broader context of partition theory and related combinatorial topics.